This is a situation involving a Christian man and a woman. I have several questions. The man is married and his wife is still living. The woman's husband has passed away. and she is now single. As friends, they became close. At times they have embraced each other. They also have touched each other inappropriately. They have not had intercourse. Is this a sin? Is it a private sin that God can forgive when the Christian goes to God in prayer and does not require a public confession before the congregation?
I don't know of a passage that divides sins into categories of public or private. See Do I have to make a public confession for sins that I mentioned to friends before I repented?
You acknowledged that the behavior was wrong when you called it "inappropriate." Paul stated, "Now concerning the things about which you wrote, it is good for a man not to touch a woman" (I Corinthians 7:1). Unmarried couples have no business touching each other sexually. It can also be classified as lewdness or sensuality. "Let us behave properly as in the day, not in carousing and drunkenness, not in sexual promiscuity and sensuality, not in strife and jealousy. But put on the Lord Jesus Christ, and make no provision for the flesh in regard to its lusts" (Romans 13:13-14).
The idea that sexual sins start with intercourse is false. "But I say to you that everyone who looks at a woman with lust for her has already committed adultery with her in his heart" (Matthew 5:28). The man is failing his marriage and his Lord. They both need to be spoken to and rescued from their sins.